Bogoliubov Theory of weakly interacting Bose gas

[barnflakes]

Please see attached file. Can anyone explain how he got the terms in the second step please? I can see the first term is with k' = k = q = 0, and I can see the next contains k ' = 0, q = 0 and k = k, but I can't see how he gets the others.

Thanks.

[barnflakes]

Sorry, I realise the attachment was missing. Here it is. I don't understand how he gets the expression in the box from the expression above it. I can get the first sum, for k = k' = q = 0 but not the other terms, does anyone know how it's done?

[guyingfei]

I am not very clear about your question. But I guess that is the mean-field solution, so the 4-operator reduce to 2-operator with some proper approximate